Correlation
between two random variables is a number between –1 and +1
0
Correlation means that there is NO
predictive connection between them
*
the
outcome of one variable tells you nothing
about the outcome of the other variable
+
Correlation means that the outcomes tend
to move in the same direction
*
outcome
of one variable higher than expected
tells you that the outcome of the other variable will tend to be higher than
expected
*
outcome
of one variable lower than expected
tells you that outcome of the other variable will tend to be lower than
expected
*
+1
correlation tells you the tendency is
almost certain
–
Correlation means that the outcomes tend
to move in opposite directions
*
outcome
of one variable higher than expected
tells you that the outcome of the other variable will tend to be lower than
expected
*
outcome
of one variable lower than expected
tells you that the outcome of the other variable will tend to be higher than
expected
*
–1
correlation tells you tendency is almost certain
Pooling
two risks (random variables; uncertain outcomes) means that each agrees to bear
half of the total of the two outcomes … each bears the average outcome. One Participant Two Participants
If
identical risks are pooled the example shows that:
*
the
expected value doesn’t change
If
the risks are also uncorrelated (0
correlation, independent):
*
the
variance (std deviation; “risk”) is lower
*
relatively
more probability winds up closer to the expected value Two
Participants
*
total outcome easier to predict
than separate outcomes
*
share of the total easier to predict than separate
outcome
* the example was identical risks but so long as they are uncorrelated, within reason, similar but non-identical risks work the same way … just assign each its expected share of the total expected value (give a quick example) … that’s part of underwriting
The
law of large numbers says that the
larger the number of uncorrelated risks, the more the probability of the
average outcome (the “pooled” outcome) gets concentrated close to the expected
value (draw a graph)
*
but
don’t forget, in the real world you usually don’t know the expected value with
perfect accuracy
The
central limit theorem says that the
larger the number of uncorrelated risks, the more the probability distribution
of the average outcome (the “pooled” outcome) looks like a normal distribution
*
the
central limit theorem can fail to be true if the initial distribution (the one
that applies to each risk separately) is too extreme … even though the law of
large numbers will still apply, the probability distribution of the average
outcome can be skewed rather than normal in these extreme cases
*
some
insurance risks are suspected to be too extreme for the central limit theorem
to apply
First,
what would cause risks to be positively correlated? Underwriting (more later) tries to fight these causes of positive
correlation (for most risk management, don’t worry about negative correlation –
the world isn’t that kind to us)
*
a
common event causes or affects more than one of the risks
§
one
conflagration burns many buildings (the old map clerk; reinsurance) – affects
frequency
§
one
hurricane damages many properties (coastal underwriting control; reinsurance) –
affects frequency and severity
§
one
accident causes many injuries and/or deaths (WTC) – frequency and severity
(double indemnity)
§
one
epidemic causes many illnesses and/or deaths (1918-19 flu; AIDS) – frequency
and maybe severity
*
a
common process influences more than one risk simultaneously
§
inflation
raises all loss costs (replacement and/or service costs) more than expected –
severity
§
litigious
attitude spreads in society faster than expected – frequency and/or severity
§
tendency
of people to use healthcare services increases faster than expected – frequency
and/or severity
What
does positive correlation do to pooling?
*
It
erases some of the risk-reduction (draw graph)
*
Not
all of it (unless the correlation goes to +1) (draw graph)
*
It
works just as if fewer risks were in the pool in the first place
A major challenge for pooling is to keep correlation at bay as much as possible
Costs of Pooling (also called “contracting costs”)
Distribution
Costs and Function – SELLING
*
Independent
agents/brokers; exclusive (tied) agents/employees; direct response channels
(now including web)
*
Obvious
function: get plenty of participants into the pool
*
Equally
important:
§
get
uncorrelated participants – to protect true pooling and ensure risk (variance)
reduction
§
get
(select) participants who didn’t just
search out the pool in order to take advantage of it (called anti-selection) – to protect the average
outcome; (don’t sell insurance to anyone who wants it too badly, instead you
want slightly reluctant participants whom your sales effort has uncovered and
convinced to join)
*
Whole
life insurance distribution costs can chew up the whole first year premium –
roughly equal in value to avoiding one “extra” death per thousand per year
*
Skilled
employees; organized rules, processes and discipline (now including artificial
intelligence technology); investigation and data collection (now including data
mining); bureaus/services
*
Obvious
function: discover and turn away participants attempting to take advantage of
the pool (i.e. fight anti-selection)
*
Equally
important:
§
assign
each participant reasonably accurately to a group with similar expected costs
and risk (classification and pricing)
§
screen
participants and the terms of pooling (the contracts) to avoid creating
incentives to behave in ways that threaten the pool (moral hazard)
*
Investigators,
decision-makers, and lawyers; employees and outside services/firms;
investigation, evaluation, negotiation, litigation, and agreement
*
Obvious
function: detect frauds and assign fair value to legitimate losses; handle the
resulting payments
*
Equally
important:
§
diminish
moral hazards (exaggeration of
losses, behavioral increase to losses – e.g. disability income claim)
§
sentinel effect – handle today’s claims in
a way that makes tomorrow’s claimants/lawyers less likely to abuse the system
§
in
a pure pooling system (more theoretical
than realistic): assign and collect the assessments to pay for the losses
*
Ex post assessment of loss costs
usually difficult/impossible
*
Ex ante assessment of “expected”
loss costs works – called “premiums”
*
Requires
expert data collection, analysis, projection and assessment skills:
underwriters and actuaries; also administration of the collection and
recordkeeping
*
Obvious
function: assess total premiums enough to cover total “expected” losses plus
required loadings
*
Equally
important: maintain reasonable enough relationship between each participant’s
premium and expected losses to avoid an anti-selection spiral
*
Subject
both to sheer error and to residual uncertainty risk – what if not enough is
assessed and collected? Can’t go back ex post
Insurance
institution capital stands behind any shortfall
*
Excess
of economic value of assets over economic value of liabilities – might not
equal the accounting book values
*
It’s
like collateral for the promised loss payments
Since no insurance institution’s capital is infinite (U.S. gov’t.?) there remains some residual uncertainty risk when insurance is used to manage risk … risk manager (maybe with agent/broker help) needs to assess that risk
*
Insurance
companies can and do become insolvent and fail to make promised loss payments
(more in later classes about mechanisms to avoid that … but they are not 100%
fail-safe)